Q119 / 160 · AP EAMCET 2019 (23 Apr Shift 1) · Mathematics

The differential equation corresponding to the family of curves \(y=e^x(A \cos x+B \sin x)\) is

  • A
    \(y^{\prime \prime}+y^{\prime}+y=0\)
  • B
    \(y^{\prime \prime}+2 y^{\prime}+2 y=0\)
  • Correct answer
    C
    \(y^{\prime \prime}-2 y^{\prime}+2 y=0\)
  • D
    \(y^{\prime \prime}-2 y^{\prime}-2 y=0\)

Solution

Given, curve \(y=e^x(A \cos x+B \sin x)\) ...(i) So, \(y^{\prime}=e^x(-A \sin x+B \cos x)+e^x(A \cos x+B \sin x)\) \(\Rightarrow \quad y^{\prime}-y=e^x(-A \sin x+B \cos x)\) ...(ii) So, \(y^{\prime}-y^{\prime}=e^x(-A \cos x-B \sin x)\) \(+e^x(-A \sin x+B \cos x)\) \(\Rightarrow \quad y^{\prime \prime}-y^{\prime}=-y+y^{\prime}-y\) [from Eqs. (i) and (ii)] \(\Rightarrow \quad y^{\prime \prime}-2 y^{\prime}+2 y=0\) Hence, option (c) is correct.

Hence, the correct answer is option: 'C'.

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